The classes of Lie-admissible algebras and power-associative algebras have been introduced by Albert in [1]}. An algebra is called Lie-admissible if the skew bilinear map \(\psi \)  related to \(\mu \)  is a Lie bracket. This is equivalent to write \(\mathcal {A}_\mu \circ \Phi ^A_{v_{Lad}}=0\) . An algebra is said to be power-associative if the subalgebra generated by any element is associative.
Over a field of characteristic 0, an algebra is power-associative if and only if it satisfies \(\mathcal {A}_{\mu }(x,x,x)=\mathcal {A}_{\mu }(x^2,x,x)=0\) 
for any \(x \in A\) . An algebra is said to be 3-power-associative if and only if it satisfies \(\mathcal {A}_{\mu }(x,x,x)=0\)  for any \(x \in A\) . This last condition is equivalent to \(\mathcal {A}_\mu \circ \Phi ^A_{v_{3Pa}}=0.\)  This implies
\(\mathcal {A}_\mu (x^2,x,x)+\mathcal {A}_\mu (x,x^2,x)+\mathcal {A}_\mu (x,x,x^2)=0.\)  But we also trivially have \(\mathcal {A}_\mu (x^2,x,x)-\mathcal {A}_\mu (x,x^2,x)+\mathcal {A}_\mu (x,x,x^2)=0.\)  We deduce \(\mathcal {A}_\mu (x,x^2,x)=0\)  and so \(\mathcal {A}_\mu (x^2,x,x)+\mathcal {A}_\mu (x,x,x^2)=0.\)  Then a 3-power-associative algebra is power-associative if and only if
\(\mathcal {A}_\mu (x^2,x,x)-\mathcal {A}_\mu (x,x,x^2)=0.\) 
So a necessary condition for a 3-power-associative to be power-associative is \(\mathcal {A}_\mu \circ \Phi _{Id-\tau _{13}}= 0.\)
